There's no need to run without a router. In fact, if you don't, then you won't be able to share the one and only public IP! You can't just patch a switch behind the modem. Then every device behind it will attempt to acquire the public IP and all but one will fail.
All you need to do is patch the router to the modem, then patch your Gigabit (1000Mbps) switch to the router's switch. Finally, patch all your Gigabit devices behind the Gigabit switch.
It doesn't matter if the modem is Gigabit anyway. No internet provider I’m aware of is likely to reach Gigabit speeds anyway, not even close. 100Mbps should be more than enough.
[modem]<-- wire -->[router]<-- wire -->[gigabit switch]<-- wire(s) -->[gigabit devices]